艾滋病及狂犬病的数学模型及其动力学分析

发布时间：2018-06-12 21:29

本文选题：Hopf分支 + 数值模拟　；参考：《兰州大学》2008年硕士论文

【摘要】： 传染病的传播模型可追述到1760年Daniel Bernoulli对天花的分析。传染病动力学的建模与研究于二十世纪中叶开始蓬勃发展,作为标志性的著作是Bailey于1957年出版的专著《数理流行病学》。近年来,国际上传染病动力学的研究进展迅速,大量的数学模型被用于分析各种各样的传染病问题,也有部分是针对诸如麻疹、流感等诸多具体疾病的模型。从模型的角度来划分,主要可以分为四类:一类是研究以常微分方程描述的流行病模型;一类是研究以偏微分方程描述的流行病模型;第三类是研究同时含有常微分方程和偏微分方程的流行病模型,这主要包括类年龄结构的流行病模型;另外一类模型为随机模型,可以在相应常微分方程的基础上增加随机考虑或利用Markov链进行Monte Carlo模拟。本文是针对艾滋病,狂犬病而建立了常微分方程模型。艾滋病是由于人体感染人类免疫缺陷病毒而引起的一种病死率极高的恶性传染病。狂犬病是由狂犬病毒引起的人兽共患烈性传染病,一旦发病病死率几乎100%。本文基于其相关的病理知识,分别对应建立了数学模型,并用动力学的知识对其进行分析,通过数值模拟,得到以下结论: 1.在病毒对健康细胞的感染率是非线性时,找出了决定系统周期解的稳定性,周期解的周期的因素。 2.通过对目前狂犬病流行现状的研究,在考虑中国实际情况的基础上,建立了新的数学模型。本模型不仅涉及到狂犬病在犬中传播,还考虑了暴露和患病犬以及携带病毒的”健康犬”使人感染狂犬病的现象。通过寻找”基本再生数”对比了捕杀、免疫、捕杀和免疫相结合三种不同策略在控制狂犬病传播中的有效性。分析和模拟结果表明三种控制狂犬病的方法中捕杀的效果最好,免疫的效果次之,捕杀和免疫相结合的效果最差。同时根据中国目前城市和农村发展不平衡的现状,提出了在城市以免疫为主,在农村采用捕杀和免疫相结合的控制狂犬病的措施,从而为中国目前控制狂犬病的流行提供了理论依据。
[Abstract]:The transmission model of infectious diseases can be traced back to Daniel Bernoulli's analysis of smallpox in 1760. The modeling and research of infectious disease dynamics began to flourish in the middle of the 20th century. As a landmark work Bailey published his monograph "Mathematical Epidemiology" in 1957. In recent years, the dynamics of infectious diseases has made rapid progress in the world. A large number of mathematical models have been used to analyze various infectious disease problems, and some models are aimed at many specific diseases such as measles, influenza and so on. From the point of view of model, it can be divided into four categories: one is to study epidemic model described by ordinary differential equation, the other is to study epidemic model described by partial differential equation. The third is to study epidemic models with both ordinary differential equations and partial differential equations, which mainly include age-like epidemic models and stochastic models. It is possible to add random consideration to the corresponding ordinary differential equations or to use Markov chains for Monte Carlo simulation. In this paper, an ordinary differential equation model is established for AIDS and rabies. AIDS is a malignant infectious disease caused by human immunodeficiency virus infection. Rabies is a zoonotic infectious disease caused by rabies virus. Based on the relevant pathological knowledge, the mathematical models are established and analyzed with the knowledge of dynamics. By numerical simulation, the following conclusions are obtained: 1. When the infection rate of virus to healthy cells is nonlinear, the factors that determine the stability of the periodic solution and the period of the periodic solution of the system are found. 2. A new mathematical model was established based on the study of the current situation of rabies in China. This model not only deals with the spread of rabies in dogs, but also takes into account the fact that exposed and diseased dogs and "healthy dogs" carrying the virus cause rabies infection in humans. By looking for "basic regeneration number", this paper compares the effectiveness of three different strategies of killing, immunization, killing and immunization in controlling rabies transmission. The results of analysis and simulation show that among the three methods of rabies control, the killing effect is the best, the immune effect is the second, and the combination of killing and immunization is the worst. At the same time, according to the current situation of unbalanced development between urban and rural areas in China, the measures of controlling rabies are put forward, which are mainly immunization in cities and combination of killing and immunization in rural areas. It provides a theoretical basis for controlling the prevalence of rabies in China.
【学位授予单位】：兰州大学
【学位级别】：硕士
【学位授予年份】：2008
【分类号】：R311

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